Number Theory Through Inquiry.

Intro -- Title page -- Contents -- 0 Introduction -- Number Theory and Mathematical Thinking -- Note on the approach and organization -- Methods of thought -- Acknowledgments -- 1 Divide and Conquer -- Divisibility in the Natural Numbers -- Definitions and examples -- Divisibility and congruence -- The Division Algorithm -- Greatest common divisors and linear Diophantine equations -- Linear Equations Through the Ages -- 2 Prime Time -- The Prime Numbers -- Fundamental Theorem of Arithmetic -- Applications of the Fundamental Theorem of Arithmetic -- The infinitude of primes -- Primes of special form -- The distribution of primes -- From Antiquity to the Internet -- 3 A Modular World -- Thinking Cyclically -- Powers and polynomials modulo n -- Linear congruences -- Systems of linear congruences: the Chinese Remainder Theorem -- A Prince and a Master -- 4 Fermat's Little Theorem and Euler's Theorem -- Abstracting the Ordinary -- Orders of an integer modulo n -- Fermat's Little Theorem -- An alternative route to Fermat's Little Theorem -- Euler's Theorem and Wilson's Theorem -- Fermat, Wilson and . . . Leibniz? -- 5 Public Key Cryptography -- Public Key Codes and RSA -- Public key codes -- Overview of RSA -- Let's decrypt -- Hard Problems -- 6 Polynomial Congruences and Primitive Roots -- Higher Order Congruences -- Lagrange's Theorem -- Primitive roots -- Euler's phi-function and sums of divisors -- Euler's phi-function is multiplicative -- Roots modulo a number -- Sophie Germain is Germane, Part I -- 7 The Golden Rule: Quadratic Reciprocity -- Quadratic Congruences -- Quadratic residues -- Gauss' Lemma and quadratic reciprocity -- Sophie Germain is germane, Part II -- 8 Pythagorean Triples, Sums of Squares, and Fermat's Last Theorem -- Congruences to Equations -- Pythagorean triples -- Sums of squares -- Pythagorean triples revisited.

Fermat's Last Theorem -- Who's Represented? -- Sums of squares -- Sums of cubes, taxicabs, and Fermat's Last Theorem -- 9 Rationals Close to Irrationals and the Pell Equation -- Diophantine Approximation and Pell Equations -- A plunge into rational approximation -- Out with the trivial -- New solutions from old -- Securing the elusive solution -- The structure of the solutions to the Pell equations -- Bovine Math -- 10 The Search for Primes -- Primality Testing -- Is it prime? -- Fermat's Little Theorem and probable primes -- AKS primality -- Record Primes -- A Mathematical Induction: The Domino Effect -- The Infinitude Of Facts -- Gauss' formula -- Another formula -- On your own -- Strong induction -- On your own -- Index -- About the Authors.

Number Theory Through Inquiry is an innovative textbook that leads students on a carefully guided discovery of introductory number theory. The book has two equally significant goals. One goal is to help students develop mathematical thinking skills, particularly, theorem-proving skills. The other goal is to help students understand some of the wonderfully rich ideas in the mathematical study of numbers. This book is appropriate for a proof transitions course, for an independent study experience, or for a course designed as an introduction to abstract mathematics. Math or related majors, future teachers, and students or adults interested in exploring mathematical ideas on their own will enjoy Number Theory Through Inquiry.Number theory is the perfect topic for an introduction-to-proofs course. Every college student is familiar with basic properties of numbers, and yet the exploration of those familiar numbers leads us to a rich landscape of ideas. Number Theory Through Inquiry contains a carefully arranged sequence of challenges that lead students to discover ideas about numbers and to discover methods of proof on their own. It is designed to be used with an instructional technique variously called guided discovery or Modified Moore Method or Inquiry Based Learning (IBL). Instructors' materials explain the instructional method. This style of instruction gives students a totally different experience compared to a standard lecture course. Here is the effect of this experience: Students learn to think independently: they learn to depend on their own reasoning to determine right from wrong; and they develop the central, important ideas of introductory number theory on their own. From that experience, they learn that they can personally create important ideas, and they develop an attitude of personal reliance and a sense that they can think effectively

about difficult problems. These goals are fundamental to the educational enterprise within and beyond mathematics.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.